Let \(N_q(g)\)  be the maximum number of rational places of function fields over \(\mathbb {F}_q\)  with genus \(g\) . The real number \(A(q)\)  defined by \(A(q):=\limsup _{g\rightarrow \infty } \frac{N_q(g)}{g}\)  is called the Ihara's constant. If \(q\)  is a square, then \(A(q)=\sqrt{q}-1\)  [1]}.
The famous Tsfaman-Vlăduţ-Zink bound (TVZ bound for short) can be found from [2]}.
